A quantitative notion of redundancy for finite frames

TL;DR

This paper introduces quantitative measures called upper and lower redundancies for finite frames in Hilbert spaces, providing a more intuitive and precise understanding of redundancy than traditional definitions.

Contribution

It defines and characterizes upper and lower redundancies based on a redundancy function, improving the conceptual and mathematical understanding of frame redundancy.

Findings

Redundancies are characterized by maximum and minimum of a redundancy function.

Range of upper and lower redundancies is fully characterized.

Redundancies satisfy all desirable properties for finite frames.

Abstract

The objective of this paper is to improve the customary definition of redundancy by providing quantitative measures in its place, which we coin upper and lower redundancies, that match better with an intuitive understanding of redundancy for finite frames in a Hilbert space. This motivates a carefully chosen list of desired properties for upper and lower redundancies. The means to achieve these properties is to consider the maximum and minimum of a redundancy function, which is interesting in itself. The redundancy function is defined on the sphere of the Hilbert space and measures the concentration of frame vectors around each point. A complete characterization of functions on the sphere which coincide with a redundancy function for some frame is given. The upper and lower redundancies obtained from this function are shown to satisfy all of the intuitively desirable properties. In addition, the range of values they assume is characterized.